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Saturday, December 09, 2006

Superultramodern Science (SS) and The Millennium Problems in Mathematics

Author: Kedar Joshi (1979- )

In this article I address 3 of the 7 millennium problems in mathematics announced by the Clay Mathematics Institute (CMI), USA. I propose solutions (not all of which are meant to be conclusive) to the problems using the ideas in Superultramodern Science (SS), which is my foremost creation. (The remaining 4 problems seem to be outside the scope of SS.) It is of utmost importance to note that the nature of the ideas and consequently of the solutions is very radical and it would take painstaking efforts to fully understand and appreciate the solutions proposed. Also it has to be considered that according to Conmathematics (Conceptual Mathematics) : the superultramdoern mathematical science, the superultramodern scientific solutions to the problems are, though apparently philosophical, in fact, mathematical. Virtually all of the 3 problems are such that they demand revolutionary changes in the current (modern/ultramodern) sciences. And SS is thought to be an appropriate change. I shall state the problems exactly as they are stated on the website of the CMI. However, the statements are the ones which are brief and not the ones that are official and descriptive. This choice is out of the revolutionary nature of the solutions which makes it senseless to consider the conventional or orthodox symbolic patterns which essentially make the (official) statements look complicated and descriptive.

1. Yang - Mills Theory The laws of quantum physics stand to the world of elementary particles in the way that Newton's laws of classical mechanics stand to the macroscopic world. Almost half a century ago, Yang and Mills introduced a remarkable new framework to describe elementary particles using structures that also occur in geometry. Quantum Yang-Mills theory is now the foundation of most of elementary particle theory, and its predictions have been tested at many experimental laboratories, but its mathematical foundation is still unclear. The successful use of Yang-Mills theory to describe the strong interactions of elementary particles depends on a subtle quantum mechanical property called the ""mass gap:"" the quantum particles have positive masses, even though the classical waves travel at the speed of light. This property has been discovered by physicists from experiment and confirmed by computer simulations, but it still has not been understood from a theoretical point of view. Progress in establishing the existence of the Yang-Mills theory and a mass gap and will require the introduction of fundamental new ideas both in physics and in mathematics.

SS solution : I suppose that light, for example, is a classical wave and photon, for example, is a quantum particle. It's an assumption in modern/ultramodern science (relativity theory) that no massive entity travels at (or above) the speed of light. From the Superultramodern Scientific perspective [in particular, the NSTP (Non - Spatial Thinking Process) theoretical perspective] space is a form of illusion, mass is bulk or quantity of matter, wave and particle are two conceptually distinct entities existing in the form of non-spatial states of consciousness/feelings. To sum up, wave -particle behaviour is an orderly governed illusion where the massive quantum particles do not really travel in space but are presented at the time of wave collapse.

2. Poincare Conjecture If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is ""simply connected,"" but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

SS solution : According to the Joshian conjecture in Superultramdoern Science (SS), that space has 3 and only 3 spatial dimensions, the concept of three dimensional sphere (and consequently Poincare conjecture itself) is absurd.

3. P vs NP Suppose that you are organizing housing accommodations for a group of four hundred university students. Space is limited and only one hundred of the students will receive places in the dormitory. To complicate matters, the Dean has provided you with a list of pairs of incompatible students, and requested that no pair from this list appear in your final choice. This is an example of what computer scientists call an NP-problem, since it is easy to check if a given choice of one hundred students proposed by a coworker is satisfactory (i.e., no pair from taken from your coworker's list also appears on the list from the Dean's office), however the task of generating such a list from scratch seems to be so hard as to be completely impractical. Indeed, the total number of ways of choosing one hundred students from the four hundred applicants is greater than the number of atoms in the known universe! Thus no future civilization could ever hope to build a supercomputer capable of solving the problem by brute force; that is, by checking every possible combination of 100 students. However, this apparent difficulty may only reflect the lack of ingenuity of your programmer. In fact, one of the outstanding problems in computer science is determining whether questions exist whose answer can be quickly checked, but which require an impossibly long time to solve by any direct procedure. Problems like the one listed above certainly seem to be of this kind, but so far no one has managed to prove that any of them really are so hard as they appear, i.e., that there really is no feasible way to generate an answer with the help of a computer. Stephen Cook and Leonid Levin formulated the P (i.e., easy to find) versus NP (i.e., easy to check) problem independently in 1971.

SS solution : According to the NSTP theory, one of the major of components of SS, all problems which, in principle, have answers are, in fact, P problems. This implication is based on the idea of the non -spatial superhuman computer that takes zero time to process information.